An Axisymmetric Steady State Vortex Ring Model

An axisymmetric steady state vortex ring model

Ruo-Qian Wanga,∗

aDepartment of Civil and Environmental Engineering, Massachusetts Institute of Technology, MA 02139, USA

Abstract Based on the solution of Atanasiu et al. (2004), a theoretical model for axisymmetric vortex flows is derived in the present study by solving the vorticity transport equation for an inviscid, incompressible fluid in cylindrical coordinates. The model can describe a variety of axisymmetric flows with particular boundary conditions at a moderately high Reynolds number. This paper shows one example: a high Reynolds number laminar vortex ring. The model can represent a family of vortex rings by specifying the modulus function using a Rayleigh distribution function. The characteristics of this vortex ring family are illustrated by numerical methods. For verification, the model results compare well with the recent direct numerical simulations (DNS) in terms of the vorticity distribution and streamline patterns, cross-sectional areas of the vortex core and bubble, and radial vorticity distribution through the vortex center. Most importantly, the asymmetry and elliptical outline of the vorticity profile are well captured. Keywords: vorticity dynamics, Norbury-Fraenkel family, Whittaker function

1. Introduction The studies of vortex rings can be traced back more than one and half centuries, when William Barton Rogers, founder of MIT, conducted the ﬁrst systematic vortex ring experiments (Rogers, 1858). Inspired by his and other

arXiv:1601.06414v1 [physics.flu-dyn] 24 Jan 2016 pioneers’ work, a series of theoretical endeavors have been made for the mathematical description of vortex rings. An early example is the famous

∗Corresponding author. Tel.:+1-617-253-6595. Email address: [email protected] (Ruo-Qian Wang)

Preprint submitted to Applied Mathematical Modeling June 11, 2021 Hill’s vortex, which assumes that vorticity is linearly proportion to radius within a spherical volume, with potential flow outside (Hill, 1894). The assumption is relatively simple, yet it is able to generate realistic-looking streamlines, and is arguably the most popular vortex ring model in applied science and engineering, e.g. Lai et al. (2013). From a different starting point, Fraenkel (1972) analyzed the vortex ring by extending the theoretical solution of a vortex filament to allow for a small finite thickness. To bridge these two models, Norbury (1973) treated the Hill’s spherical vortex and Fraenkel’s thin ring as two asymptotic members of a series of generalized vortex rings. He then numerically determined a range of intermediate rings, now referred to as the Norbury-Fraenkel (NF) vortex ring family. The NF family delivers more accurate streamlines and more precise vortex ring outlines. However, its linear distribution of vorticity is still unrealistic (Danaila & Hélie,2008). Recently, a solution for viscous vortex rings at low Reynolds numbers (Re) was obtained by Kaplanski & Rudi (1999). They derived a generalized solution to the diffusing viscous vortex ring, by directly solving the axisymmetric vorticity-stream function equations without the nonlinear convection terms. Their solution delineates a donut shape outline and a more realistic Gaussian distribution of vorticity in the radial direction. Kaplanski et al. (2009) extended the solution to turbulence by adopting an effective turbulent viscosity. However, recently Danaila & Hélie(2008) conducted Direct Numerical Simulations (DNS) and reported elliptical cross-sections and radial asymmetry for the vorticity distribution, which both Norbury-Fraenkel and Kaplanski-Rudi models are unable to capture. Realizing the issue, Ka- planski et al. (2012) added two adjustable parameters to allow an elliptical cross-section, but the radial asymmetry of the vorticity distribution is still amiss due to their symmetric Gaussian distribution. Laboratory experiments and numerical simulations have also shed light on vortex ring dynamics as reviewed by Lim & Nickels (1995) and Shariff & Leonard (1992). In particular, Gharib et al. (1998) performed experiments in which a piston generated a non-buoyant vortex puff. They observed that if the aspect ratio of piston stroke length L to diameter D was less than about four, the generated vorticity could be incorporated into the head vortex, whereas for larger aspect ratios a trailing stem occurs. The phenomenon is now referred to as the “pinch-off”, and the critical aspect ratio is called the “formation number”. Because the trailing stem stops supplying vorticity to the head vortex after the pinch-off, the “saturated” head vortex ring in the post-formation stage is relatively stable, and should resemble the steady

2 state situation in Hill’s vortex. The experimental results can advance the state-of-the-art by enabling a meaningful comparison with idealized theoretical models and numerical simulations or experiments with a saturated head vortex ring. An example can be found in Danaila & Hélie(2008), and the present study is also initiated in the same spirit. The best theoretical model would be the analytical solution to the Navier- Stokes equations, which accurately depicts the dynamics of the flow but is difficult to derive. As a compromise, a theoretical model could be built on the solution to the Euler equations, which ignores the viscous effect but captures the more important non-linear dynamics of the flow. In the present case, we focus on the steady state axisymmetric Euler equation, the target of which is to solve an elliptical second order partial differential equation. This equation is a particular form of the Grad-Shafranov equation (Shafranov, 1958; Grad & Rubin, 1958), which is related to the magnetostatic equilibrium in a perfectly conducting fluid and is well known in magnetohydrodynamics (MHD). Specifying different forms of the involved arbitrary functions, plasma physicists have derived a series of analytical solutions to this equation, e.g. the simplest solution of the Solov’ev equilibrium (Solov’ev, 1968), the Herrnegger-Maschke solution (Herrnegger, 1972; Maschke, 1973), and the recent more general solution by Atanasiu et al. (2004) (hereafter referred to as AGLM). It can be shown that the Hill’s spherical vortex corresponds to the Solov’ev equilibrium, yet no counterparts of the more advanced solutions have been explored for vortex dynamics. This encourages us to take advan- tage of them, especially the AGLM, to reach a more accurate model of vortex rings. Beyond that, a higher accuracy model can also meet the industrial de- mand to improve the description of vortex structures to replace the aged Hill’s vortex, e.g. Lai et al. (2013). In particular, a more realistic vorticity distribution may lay a better foundation to address the particle-vorticity interactions, which motivates us to make the following study. As mentioned earlier, existing models have improved progressively to describe more detailed features of real vortex rings. However, an outstanding issue is the asymmetry of the vorticity distribution. The present paper provides an alternative analysis that is capable of incorporating the asymmetry, by deriving a theoretical model for high Re laminar vortex rings based on the solution to the axisymmetric inviscid vortex flow. The governing equations are first presented in section 2 and shown to relate to the Grad-Shafranov equation. The alternative vortex ring solution is shown in section 3, and its properties are derived in section 4. Then, the model is compared to existing

3 simulation results in section 5. Summary and conclusions are given in section 6.

2. A particular solution We adopt the cylindrical coordinate system (r, φ, x), where x is the local longitudinal coordinate. The incompressible axisymmetric vorticity transport equations can be stated as (Fukumoto & Kaplanski, 2008)

∂ξ ∂ ∂ µ ∂2ξ ∂2ξ 1 ∂ξ ξ + (vξ) + (uξ) = + + − (1) ∂t ∂r ∂x ρ ∂x2 ∂r2 r ∂r r2 where t is time, ξ is the vorticity, u and v are velocity components in x and r directions respectively, µ is the dynamic viscosity, and ρ is the density. Introducing the Stokes stream function ψ, the velocity components can then be expressed as 1 ∂ψ 1 ∂ψ u = − + U , v = , (2) r ∂r x r ∂x where Ux is the translational velocity of the vortex centroid. The vorticity can be deﬁned by the stream function as

∂2ψ ∂2ψ 1 ∂ψ + − = −rξ, (3) ∂r2 ∂x2 r ∂r which provides the closure to solve (1). Equations (1)-(3) can therefore be applied to any axisymmetric ﬂows in theory. Substituting (2) into (1) and normalizing the variables by a length scale l and a velocity scale U, i.e.

∗ ∗ ∗ ∗ 2 ∗ −1 ∗ u = Uu , v = Uv , r = lr , x − Uxt = lx , ψ = l Uψ , ξ = Ul ξ , (4) a non-dimensionalized form of (1) can be derived as

∂ξ∗ ∂ ξ∗ ∂ψ ∂ ξ∗ ∂ψ ∂ − U + − + (U ξ∗) x ∂x∗ ∂r∗ r∗ ∂x∗ ∂x∗ r∗ ∂r∗ ∂x∗ x 1 ∂2ξ∗ ∂2ξ∗ 1 ∂ξ∗ ξ∗ = + + − , (5) Re ∂x∗2 ∂r∗2 r∗ ∂r∗ r∗2

4 where Re = ρUl/µ. U and l can be specified for a particular application. Note that if Re 1, the nonlinear terms on the LHS of (5) can be neglected, leading to the theoretical solution obtained earlier by Kaplanski & Rudi (1999). For Re 1 but before transition into turbulence, the viscous effect (RHS of equation (5)) can be ignored in the high Re laminar flows. Assuming a steady translation such that Ux is constant, and dropping the * sign for brevity, then ∂ ξ ∂ψ ∂ ξ ∂ψ − = 0. (6) ∂r r ∂x ∂x r ∂r Using ξ = rf(ψ) (Shariff & Leonard, 1992) and substituting into (3), we get

∂2ψ ∂2ψ 1 ∂ψ + − = −r2f (ψ) . (7) ∂r2 ∂x2 r ∂r As mentioned earlier, equation (7) is a particular form of the Grad-Shafranov equation: ∂2ψ ∂2ψ 1 ∂ψ + − = −r2f (ψ) + g (ψ) , (8) ∂r2 ∂x2 r ∂r with the arbitrary function g(ψ) = 0. The simplest particular solution to (8) is the Solov’ev equilibrium solution, which assumes f(ψ) = A, g(ψ) = B (9) where A and B are constants. If B = 0, this reduces to the Hill’s spherical vortex solution (Hill, 1894), which can be solved in spherical coordinates by separation of variables, while confining the vorticity within the spherical vortex boundary. Although Hill’s streamlines are similar to experimental and numerical observations, the vorticity distribution is too simplified and, as will be further discussed, differs significantly from observations (Danaila & Hélie,2008). In MHD, more solutions are available and have the potential to improve the accuracy over the simplest Hill’s spherical vortex. One example is the particular solution with f(ψ) = Aψ and g(ψ) = Bψ, which is known as the Herrnegger-Maschke solution (Herrnegger, 1972; Maschke, 1973). In addition, a more general solution of AGLM by Atanasiu et al. (2004) covers both

5 situations above, assuming a series expansion of f(ψ) and g(ψ), i.e.

f(ψ) = Aψ + B, g(ψ) = Cψ + D, (10) where A, B, C and D are constants. The AGLM solution inspires us to generate a counterpart model of vortex rings as a meaningful extension to the Hill’s spherical vortex. If C = D = 0, equation (7) becomes

∂2ψ ∂2ψ 1 ∂ψ + − = −Ar2ψ − Br2. (11) ∂r2 ∂x2 r ∂r Thus, the Hill’s vortex solution can be viewed as the zeroth order approximation, A = 0, which linearizes the governing equations for solvability. Note that higher order series expansions can be used in the source term of (7), which involves the nonlinear term and no explicit solution is available so far. Recalling that ξ = rf(ψ), the improvement of (10) over (9) is the manner in which the vorticity ξ correlates with the stream function ψ. As discussed earlier, Hill’s model delivers realistic-looking stream lines, and the linear vorticity distribution is also advantageous for the handling of the viscous terms in (5) and thus becomes the only explicit solution to the Beltrami flow (Wang, 1991). However, the vorticity distribution is unrealistic. The present solution of (10) assumes that the vorticity distribution is the first radial moment of the stream function, and by doing so directly correlates the vortex and flow structures. It will be shown later that an improved matching of the stream function and vorticity distribution can be obtained in this manner. Furthermore, the correlation better reflects the physics that the vorticity is transported by the flow structure. In the following, we will show how the explicit exact solution can be sought with this first order approximation, which is similar to Atanasiu et al. (2004) but with improvements in details. The inhomogeneous nature of (11) leads to a solution consisting of two parts, i.e. ψ = ψo + ψno, where ψo and ψno are the homogeneous and inhomogeneous parts, respectively. We shall obtain a general solution for the homogeneous problem first, and then complete the solution with a particular inhomogeneous solution. Taking the Fourier transform of the homogeneous problem with respect to x, we have ∂2ψˆ 1 ∂ψˆ − + (Ar2 − λ2)ψˆ = 0, (12) ∂r2 r ∂r

6 where the Fourier transform is deﬁned by Z ∞ ˆ −iλx ψ = ψoe dx. (13) −∞

Equation (12) can be simpliﬁed by introducing η = αr2

∂2ψˆ 1 λ2 + − − ψˆ = 0, (14) ∂η2 4 4αη which has a solution of

ˆ 2 2 ψ = Cλ1M −λ2 1 (αr ) + Cλ2W −λ2 1 (αr ) (15) 4α , 2 4α , 2 where Mk,m(x) and Wk,m(x) are the Whittaker M and W functions, i√is the imaginary unit,√ and λ, Cλ1,Cλ2 are constants to be determined, α = Ai if A > 0 and α = −A if A < 0 (Polyanin et al., 2003). The solution in AGLM is also consistent with the present result but has a much lengthy form. In other words, the present solution is more compact and succinct. When A > 0, the solution can also be expressed as: √ √ 2 2 ! 2 2 ! ˆ λ Ar λ Ar ψ = Cλ1F0 √ , + Cλ2G0 √ , , (16) 4 A 2 4 A 2 where FL(η, x) and GL(η, x) are the Coulomb wave functions of the ﬁrst and the second kind. This solution is consistent with the Herrnegger-Maschke solution. Subsequently, the stream function and vorticity can be recovered by taking the inverse Fourier transform as Z ∞ h 2 2 i 2 2 ψo = Cλ1M− λ , 1 (αr ) + Cλ2W− λ , 1 (αr ) [cos(λx) − i sin(λx)]dλ. −∞ 4α 2 4α 2 (17) The inhomogeneous solution ψno should satisfy the equation of ∂2ψ ∂2ψ 1 ∂ψ no + no − no = −Ar2ψ − Br2. (18) ∂r2 ∂x2 r ∂r no

The simplest particular solution is ψno = −B/A. To summarize, the solution

7 to the reduced Grad-Shafranov equation is Z ∞ h 2 2 i 2 2 ψ = Cλ1M− λ , 1 (αr ) + Cλ2W− λ , 1 (αr ) [cos(λx)−i sin(λx)]dλ−B/A, −∞ 4α 2 4α 2 (19) and Z ∞ h 2 2 i 2 2 ξ = Ar Cλ1M− λ , 1 (αr ) + Cλ2W− λ , 1 (αr ) [cos(λx) − i sin(λx)]dλ. −∞ 4α 2 4α 2 (20) In theory, (19) and (20) apply to any axisymmetric inviscid ﬂow in steady state (i.e. with constant translation Ux) so long as the ﬁrst order approximation in (10) is adequate. We now develop a theoretical model of the high Re laminar vortex ring using these solutions together with the necessary boundary conditions.

3. Vortex ring model We seek an axisymmetric vortex ring solution with a ﬁnite boundary that is also symmetric w.r.t. the plane x = 0, i.e.

ψ = const. (r = 0), (21a) ∂ψ = 0 (x = 0), (21b) ∂x ψ = const. (at Ω), (21c) where the boundary of Ω is toroidal in the 3-dimensional domain. Ω cannot be determined a priori. Its boundary can be deﬁned by the smallest non-zero root of r in equation ξ(r, x0) = 0 at a given longitudinal position x0. Here, we only consider the case of A > 0, because A < 0 gives rise to a monotonic increase of the integral in the r direction and thus an inﬁnite boundary extent. In this case, boundary condition (21a) requires that the axis of symmetry is a streamline, and without loss of generality, B/A is chosen to be the streamfunction value. Because W λ2 1 (0) 6= 0, we have to − 4α , 2 set Cλ2 = 0 to prevent the stream function from varying at r = 0. The boundary condition (21b) comes from the observation that the vorticity distribution typically peaks at the center, and the vortex shape is nearly symmetrical with respect to the plane of x = 0. Note also that the real part 2 of the Whittaker M function M λ2 1 (αr ) is not integrable, and thus Cλ1 has − 4α , 2

8 to be a pure imaginary number. Hence, the vortex ring solution can then be expressed as Z ∞ √ 2 ψ = iC(λ)M 2 ( Ar i) cos(λx)dλ + B/A (22) λ√ i , 1 0 4 A 2 Z ∞ √ 2 ξ = Ar iC(λ)M 2 ( Ar i) cos(λx)dλ (23) λ√ i , 1 0 4 A 2 where C(λ) is a real function that can be viewed as the modulus function. Because the integral is symmetric with respect to λ = 0, the integral limits are constrained to the positive range. C(λ) needs to be resolved in order to obtain the solution. Mathematically, if C(0) 6= 0, the equation is reduced to a parallel shearing flow, which has no finite boundary in the x direction; and if C(∞) 6= 0, the integral is not finite. Therefore, for a closed boundary and for integrability, C(λ) has to vanish as λ → 0 or ∞. Physically, C(λ) can be seen as a spectrum of vortex length scales, which has reached a self-balanced state after the vortex ring formation. Hence, it is reasonable to assume that C(λ) is continuous and smooth. Although many functions are able to satisfy the conditions mentioned above, the simplest candidate would be the Rayleigh distribution function, i.e. 2 λ − λ C(λ) = e 2σ2 , (24) σ2 where σ is a constant. Taking A = 1 and B = 0, we reach a normalized form of equation (22), i.e.

Z ∞ 2 λ − λ 2 2 2σ2 2 ψ = i 2 e M λ i, 1 (l r i) cos(λlx)dλ. (25) 0 σ 4 2 Accordingly, the vorticity distribution becomes

Z ∞ 2 λ − λ 2 2 2σ2 2 ξ = r i 2 e M λ i, 1 (l r i) cos(λlx)dλ. (26) 0 σ 4 2 For illustration, two typical vortex ring patterns are shown in ﬁgures 1 and 2. Note that the pattern has been further normalized by the length scale l so that the x extent of the vortex ring is within the range from -1 to 1. Comparing ﬁgures 1 and 2, the aspect ratio, β = rmax/zmax, is greater with a greater σ, and the boundary is more rectangular.

9 Generally, equations (25) and (26) depict a family of new vortex ring models. In the following, we obtain the properties of the present model by numerical methods.

4. Properties of the vortex ring family The current model of equation (25) has only one free parameter, σ, by which a family of vortex rings can be modeled. The basic properties of this model can be described by a series of integrals of the vortex ring dynamics including the circulation (Γ), the hydrodynamic impulse (I), and the energy (E), which are deﬁned as below:

Z ∞ Z ∞ Z ∞ Z ∞ Z ∞ Z ∞ Γ = ξdxdr, I = π r2ξdxdr, E = π ξψdxdr. 0 −∞ 0 −∞ 0 −∞ (27) By varying σ, we can derive the variation of β, Γ, I, and E, which are shown in figure 3. The aspect ratio and all the integrals generally increase with larger σ. Before σ = 0.7, the aspect ratio β is increasing almost linearly. Afterwards it increases dramatically and approaches infinity at σ =∼ 0.78. The integrals increase much more quickly than β and also approaches infinity at the same place. Note that when σ = 0, the r extent approaches zero (or l → ∞), and thus the vortex ring is reduced to an infinite line. As a result, all the integrals also vanish. A useful parameter in vortex ring dynamics is the propagation velocity, Ux, which is defined earlier in this paper. Theoretically, it can be estimated by the following general equation (Helmholtz, 1858; Lamb, 1932),

R ∞ R ∞ ψ − 6x ∂ψ ξdxdr U = 0 −∞ ∂x . (28) x R ∞ R ∞ 2 0 −∞ r ξdxdr

It is difficult to derive an explicit expression for Ux. Hence, we use a numerical method to compute Ux, which is shown in figure 4. From this figure, the propagation velocity Ux decreases with larger σ, but the slope of the decrease is much reduced when σ is near ∼ 0.78. At σ → 0, Ux → ∞. Note that the results in figure 4 are normalized. The absolute values depend on the scaling. The normalized radius, lR (defined later in section 5), can be derived as a function of σ (figure 5) and its relationship to the translation velocity Ux can be found in figure 6. We can find that the radius increases with σ

10 exponentially and the translation velocity decrease with the radius quickly at small radius and slowly later. Also note that the eﬀect of A 6= 1 can be incorporated by a correction to the aspect ratio β and the length scale l. If B 6= 0, the aspect ratio of the streamlines remains, but the vorticity and other integrals will be increased accordingly.

5. Comparison to numerical simulations The present solution takes advantages that at high Re laminar flow, the viscous effects are present but negligible. This enables us to make a meaningful comparison with corresponding direct numerical simulations in this range. To examine the properties of vortex rings, Danaila & Hélie(2008) performed axisymmetric Direct Numerical Simulations of a piston generated vortex ring and compared the results with the Norbury-Fraenkel family and Kaplanski and Rudi’s analytical solutions. Their simulations are based on the formation number of L/D = 4 which is close to that of a saturated vortex ring. The comparison included the streamlines and vorticity distribution at tU0/D = 30, which are shown in figure 8(a) − (f). To quantify the characteristics of the vortex ring using the present ap- proach, the parameter, σ, has to be specified. By close examinination, the aspect ratio of DH is β = 1.23, which corresponds to σ = 0.585 in the present model from figure 3. Furthermore, the normalized standard form should be adjusted in its magnitude to compare with the simulation results, with C(λ) scaled by a factor of 0.65 and l = 0.94. Using this configuration, we can visualize the three dimensional structure of the vortex ring in figure 7. The vortex ring has a torus shape with a deformed cross-section. The outside of the vortex ring is narrower than the inner side. The toroid is also sliced to show the cross-section, where the vorticity distribution can be seen. In the following, we draw a series of comparisons to show that the present results agree well with the numerical simulations. Overall, figure 8 shows that the present solution provides a better agreement than the other peer models. A close match of streamlines and vorticity distributions is obtained as shown in figure 8 (g) and (h). The streamlines of figure 8 (b), (d), (f) and (h) illustrate that all models essentially share similar characteristics, but the present solution in (h) achieves better matching.

11 More notably, the vorticity distribution comparison demonstrates a signifi- cant improvement. As discussed before, the Norbury-Fraenkel’s family has a major shortcoming due to the unrealistic linear vorticity distribution. The Kaplanski-Rodi’s model improves with a continuous vorticity distribution at the boundaries, but the vorticity contours dictate a circular shape which is also unrealistic. The present solution better depicts the asymmetry of the vorticity distribution by accommodating multiple harmonics. The small discrepancy at the boundary might be attributed to the absence of viscous effects. With the present solution, the boundaries of the vortex cores and vortex bubbles are also found to better match the numerical simulations by Danaila & Hélie(2008). The isocontour of 5% of the maximum vorticity is taken as the vortex core boundary shown in figure 9(a). The present solution shares similar characteristics of a sharper top and flatter bottom with the numerical results. In addition, the outline of the vortex bubble where the stream function decays to zero is shown in figure 9(b). With the appropriate σ, the present solution has the same aspect ratio as DH, but underestimates slightly the r and the x extents, which may again be due to the fact that the viscous effect is neglected. Although the cross-sectional area is underestimated by 18%, this is a considerable improvement over the peer models in which ∼ 40% or more underestimation is common. The radial distribution of vorticity through the vortex center is shown in figure 10. The numerical results show a smooth peak profile, with a long tail expanding to the axis of symmetry and a steep slope at the outside. Quan- titative comparison reveals that the Norbury-Fraenkel family has an unrealistic linear profile; while the Kaplanski-Rudi model features a symmetrical Gaussian distribution which by definition cannot reproduce the asymmetry observed in the numerical results. Besides the good comparison with Danaila & Hélie(2008), similar qualita- tive agreement in vorticity distribution contours is also obtained with observations reported in the literature, e.g. Zhao et al. (2000). However, for quan- titative comparison, to the authors’ knowledge, only Mohseni et al. (2001) (MRC) and Archer et al. (2008) (ATC) offered high quality data that can be compared alongside Danaila & Hélie(2008) (DH). Both MRC and ATC also adopted Direct Numerical Simulations to reproduce the vortex ring flows similar to DH. Specifically, MRC focused on the pinch-off process. The radial vorticity distributions through the vortex ring center at different times are shown as MRC1∼MRC4 in figure 11. In contrast, ATC simulated the

12 transition of vortex rings from laminar to turbulent, and only a snap shot of the radial vorticity distribution in the laminar range for both thin and thick cores were reported. They are shown as ATC1 and ATC2 respectively in figure 11. The maximum value, ξmax, and the peak radius, lR, can be extracted directly from MRC and ATC. They are used as the vorticity and length scales for normalization of their reported data. Meanwhile, we show the present solution in figure 10 using the best fitting coefficient σ = 0.585 with DH, but with the output of the results normalized by ξmax and lR. In figure 11, clearly the present solution compares well with the numerical results by MRC and ATC (the abbreviations and their corresponding flow conditions can be found in table 1). Specifically, MRC reported their vorticity distributions as a time series of a specific simulation. In their case, the distribution was narrower in earlier time when the vortex was still developing, and became wider later on when Ux approached constant. The latter is closer to our steady state assumption in the derivation of (6) (see the constant slope of the vortex center position in figure 1 of MRC). Strictly speaking the present solution is only appropriate to be compared with the steady state MRC4, where an excellent comparison is found. Note that in MRC2, a secondary peak can be identified near the axis of symmetry, reflecting the merging trailing stem. ATC observed in the late laminar phase before turbulent onset that “the vorticity profile across the core region relaxes towards a new equilibrium state, when the axisymmetric inviscid ideal is solely a function of the stream function ψ; the distribution becomes skewed, decreasing faster toward the bubble edge than the ring centre.” This observation is consistent with the present model development, and validates our comparison exercise. Of the two cases in ATC, one was with a relatively saturated state of development shown as ATC1, and the other a thin core vortex ring with a much narrower peak that had not yet reached the saturated phase shown as ATC2. Therefore, it’s more appropriate to compare with ATC1 instead of ATC2. As expected, the present model, which is derived with the steady state assumption, matches well the former. Finally, DH is also normalized and shown in figure 11 for comparison. To summarize, the present solution is able to match closely the DNS results of laminar vortex rings with high Re reported in the literature when the post-formation stage is approached, which covers a wide range of Re = 1400 − 5500. Recently, Zhang & Danaila (2013) derived a family of vortex rings with varying levels of saturation. By examining the elliptical boundaries, they

13 proved the existence of the elliptical vortex rings. Using numerical methods, they offered data on the stream function distribution in the radial direction. Corresponding to their most saturated situation, a comparison is drawn in figure 12 after normalization with the radius and the value of the maximum of ψ, i.e. lR1 and ψmax. Although the present solution is relatively narrower, the shapes of these profiles agree well with the results from Zhang & Danaila (2013). The slight discrepancy may be due to the fact that there is no superimposed potential flow in the present solution. Although the current paper is targeted to model a high Reynolds number vortex ring, it can also be applied to truly inviscid vortex rings, which are currently investigated theoretically and experimentally in superfluid and atomic Bose-Einstein condensates, as reviewed for example by Barenghi & Donnelly (2009).

6. Summary and conclusions The present study proposes a theoretical model of a family of steady state vortex rings. First, a solution to the Grad-Shafranov equation from magnetohydrodynamics is derived, which is more succinct and compact compared with the existing solution by Atanasiu et al. (2004). Using the analogy between the equations of hydrodynamics and magnetohydrodynamics, a theoretical model to high Re laminar vortex rings is developed based on this solution, which can satisfy the D-shaped boundary conditions that are com- monly used in magnetohydrodynamics and suitable for vortex rings. A new family of theoretical vortex rings is derived based on this model with a free parameter σ. A numerical method is then used to calculate the properties of this family, including aspect ratio, circulation, impulse, energy, and propagation velocity. By choosing σ = 0.585, a good match is obtained comparing the results from the present model and direct numerical simulations. The present model is found to be superior to the Norbury-Fraenkel family by virtue of its improved accuracy in the vorticity distribution and streamline pattern. Most importantly, the asymmetry and elliptical outline of the vorticity proﬁle are well captured. Further comparison is performed with the normalized vorticity and stream function distributions reported in the literature and the present solution. Again, close agreement is obtained. The improved matching with the present solution can yield better assessment and models for engineering applications.

14 7. Acknowledgement This research was supported by the National Research Foundation Sin- gapore through the Singapore MIT Alliance for Research and Technology’s CENSAM IRG research programme. The authors would like to thank the anonymous reviewers for comments to improve this manuscript.

References

Archer, P. J., Thomas, T. G. & Coleman, G. N. 2008 Direct numerical simulation of vortex ring evolution from the laminar to the early turbulent regime. Journal of Fluid Mechanics 598, 201–226.

Atanasiu, C. V., Gnter, S., Lackner, K. & Miron, I. G. 2004 Analyt- ical solutions to the GradShafranov equation. Physics of Plasmas 11 (7), 3510.

Barenghi, C. & Donnelly, R. 2009 Vortex rings in classical and quantum systems. Fluid Dynamics Research 41 (5), 051401.

Danaila, I. & Helie,´ J. 2008 Numerical simulation of the postformation evolution of a laminar vortex ring. Physics of Fluids 20, 073602.

Fraenkel, L. 1972 Examples of steady vortex rings of small cross-section in an ideal ﬂuid. Journal of Fluid Mechanics 51, 119–135.

Fukumoto, Y. & Kaplanski, F. 2008 Global time evolution of an axisymmetric vortex ring at low Reynolds numbers. Physics of Fluids 20 (5), 053103.

Gharib, M., Rambod, E. & Shariff, K. 1998 A universal time scale for vortex ring formation. Journal of Fluid Mechanics 360 (1), 121.

Grad, H. & Rubin, H. 1958 Hydromagnetic equilibria and force-free ﬁelds. In Proceedings of the Second United Nations International Conference on the Peaceful Uses of Atomic Energy, , vol. 31, pp. 190–197. Geneva: United Nations, Geneva.

Helmholtz, H. 1858 On Integrals of the Hydrodynamical Equations: Which Express Vortex-motion. Crelle’s J.

15 Herrnegger, F. 1972 On the equilibrium and stability of the belt pinch. In Proceedings of the Fifth European Conference on Controlled Fusion and Plasma Physics, , vol. I, p. 26. Grenoble.

Hill, M. J. M. 1894 On a spherical vortex. Proceedings of the Royal Society of London 55 (331-335), 219–224.

Kaplanski, F., Fukumoto, Y. & Rudi, Y. 2012 Reynolds-number eﬀect on vortex ring evolution in a viscous ﬂuid. Physics of Fluids 24 (3), 033101.

Kaplanski, F. & Rudi, U. 1999 Dynamics of a viscous vortex ring. Inter- national Journal of Fluid Mechanics Research 26 (5-6).

Kaplanski, F., Sazhin, S. S., Fukumoto, Y., Begg, S. & Heikal, M. 2009 A generalized vortex ring model. Journal of Fluid Mechanics 622 (1), 233–258.

Lai, A. C. H., Zhao, B., Law, A. W.-K. & Adams, E. E. 2013 Two- phase modeling of sediment clouds. Environmental Fluid Mechanics .

Lamb, H. 1932 Hydrodynamics. Cambridge University Press.

Lim, T. & Nickels, T. 1995 Vortex rings. Fluid Mechanics and Its Appli- cations 30, 95–95.

Maschke, E. 1973 Exact solutions of the mhd equilibrium equation for a toroidal plasma. Plasma Physics 15 (6), 535.

Mohseni, K., Ran, H. & Colonius, T. 2001 Numerical experiments on vortex ring formation. Journal of Fluid Mechanics 430, 267–282.

Norbury, J. 1973 A family of steady vortex rings. Journal of Fluid Me- chanics 57 (03), 417.

Polyanin, A. D. a. D., Zatsev, V. F. & Zatsev, V. F. 2003 Handbook of Exact Solutions for Ordinary Diﬀerential Equations, Second Edition (Google eBook), , vol. 2. CRC Press.

Rogers, W. B. 1858 On the formation of rotating rings by air and liquids under certain conditions of discharge. E. Hayes.

16 Shafranov, V. 1958 On magnetohydrodynamical equilibrium conﬁgura- tions. Soviet Phys. JETP 6, 545–554.

Shariff, K. & Leonard, A. 1992 Vortex rings. Annual Review of Fluid Mechanics 24 (1), 235–279.

Solov’ev, L. S. 1968 The theory of hydromagnetic stability of toroidal plasma conﬁgurations. JETP 26, 400–407.

Wang, C. Y. 1991 Exact Solutions of the Steady-State Navier-Stokes Equa- tions. Annual Review of Fluid Mechanics 23 (1), 159–177.

Zhang, Y. & Danaila, I. 2013 Existence and numerical modelling of vortex rings with elliptic boundaries. Applied Mathematical Modelling 37 (7), 4809–4824.

Zhao, W., H, F. S. & Mongeau, L. G. 2000 Eﬀects of trailing jet insta- bility on vortex ring formation. Phys. Fluids 12, 589–596.

17 1 1

0.5 0.5

0 0 x/l x/l

−0.5 −0.5

−1 −1

−0.5 0 0.5 −1 −0.5 0 0.5 1 r/l r/l

Figure 1: Streamlines for σ = Figure 2: Streamlines for σ = 0.6 and β = 0.2 and β = 0.39 1.32

300 5 β Γ 250 4 I E 200 3 x 150 U 2 100

1 50

0 0 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 σ σ Figure 3: The aspect ratio, β, the Figure 4: The variation of the prop- circulation, Γ, the impulse I and the agation velocity, Ux, against the free energy E against the free parameter, parameter, σ σ

18 Figure 5: The variation of vortex Figure 6: The variation of the prop- ring radius, lR, against the free pa- agation velocity, Ux, against the ra- rameter, σ dius, lR

Figure 7: Three-dimensional vortex ring structure of vorticity.

19 Figure 8: Comparison of the ideal models of Norbury-Fraenkel, Kaplanski- Rudi, and the present solution with the direct numerical simulations byDanaila & Hélie(2008). The upper row is the vorticity distribution and the lower is the streamline. (a)-(f) modified from Danaila & Hélie(2008).

1.5 (b) 1.5 (a)

1 1

simulation by DH r

r Present model simulation by DH 0.5 0.5 present model

0 10.5 11 11.5 12 12.5 0 x 10.5 11 11.5 12 12.5 x Figure 9: Comparison of boundary contours with DH: (a) the vortex core and (b) vortex bubble.

MRC1 1 Simulations of DH MRC2 5 Norbury−Fraenkel MRC3 Kaplanski−Rudi 0.8 MRC4 Present model ATC1 4 ATC2 0.6 DH max ξ / Present ξ model ξ 3 0.4

2 0.2

0 1 0 0.5 1 1.5 2 r/l R

0 0 0.2 0.4 0.6 0.8 1 1.2 Figure 11: Radial distribution of r vorticity through the vortex cen- Figure 10: Radial distribution of ter: comparison to diﬀerent nu- vorticity through the vortex cen- merical simulations in literature. ter: comparison to diﬀerent vortex Legend entries can be found in ta- ring models ble 1.

1.4 present model 1.2 Zhang and Danaila (2013) 1

0.8 max ψ /

ψ 0.6

0.4

0.2

0 0 0.5 1 1.5 r/l R1 Figure 12: Radial distribution of the stream function through the vortex center compared to Zhang & Danaila (2013).

21 Table 1: Selected numerical results in the literature

Abbreviation Literature Re x/D MRC1 Mohseni et al.(2001) 3800 3 MRC2 5.8 MRC3 12.4 MRC4 19.9 ATC1 Archer et al.(2008) 5500 - ATC2 - DH Danaila and H´elie(2008) 1400 11.35